{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R375",
  "content_hash": null,
  "slug": "heilbronn10-claim-global-frontier-open",
  "type": "claim",
  "title": "The best-known ten-point area is 0.0465374195825..., with global optimality open",
  "summary": "The exact Comellas–Yebra construction proves Δ₁₀ ≥ 0.04653741958254177256.... No checked source proves a matching unrestricted upper bound, so the exact value of Δ₁₀ remains open.",
  "relevance": "For Exact ten-point Heilbronn number in the unit square, record heilbronn10-claim-global-frontier-open (“The best-known ten-point area is 0.0465374195825..., with global optimality open”) records a bound, answer, status fact, or structural consequence. The record states: The exact Comellas–Yebra construction proves Δ₁₀ ≥ 0.04653741958254177256....",
  "relevance_source": "recorded",
  "body": "Sudermann-Merx, arXiv:2603.11107v2, gives globally certified solutions for \\(n\\le9\\). Section 7 says that extending certification to \\(n=10\\) remains open and will require substantial computation. Appendix A, Table 11 gives the exact Comellas–Yebra ten-point construction and identifies it as best known.\n\nMonji, Modir, and Kocuk, arXiv:2512.14505v1, also leave \\(n=10\\) open. Their reported ten-point run uses a restricted model with two unproved assumptions: exactly two points on each edge and \\(y_5\\le1/2\\). The solver reached no matching upper bound within one day.\n\nThe sources therefore support an exact best-known construction and an open unrestricted upper-bound problem. The two structural assumptions belong to the Monji-Modir-Kocuk restricted computation.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the published certification status of the unrestricted ten-point Heilbronn triangle problem in the unit square as audited on 2026-07-28"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2603.11107",
      "locator": "Sudermann-Merx, arXiv:2603.11107v2, Section 1.4 on p. 3, Section 7 on p. 17, and Appendix A/Table 11 on p. 19; Monji-Modir-Kocuk, arXiv:2512.14505v1, Conjecture 1 and final n=10 experiment"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2603.11107",
    "locator": "Sudermann-Merx, arXiv:2603.11107v2, Section 1.4 on p. 3, Section 7 on p. 17, and Appendix A/Table 11 on p. 19; Monji-Modir-Kocuk, arXiv:2512.14505v1, Conjecture 1 and final n=10 experiment"
  },
  "relations": [
    {
      "slug": "R374",
      "title": "The exact Comellas–Yebra construction has minimum area 0.0465374195825...",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R371",
      "title": "Audit the live record, primary sources, and current arXiv frontier",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R373",
      "title": "Run the unrestricted Sudermann-Merx global model at n = 10",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "heilbronn-square-ten",
      "title": "heilbronn square ten",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}